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Surface Tensions, Surface Potentials, and the Hofmeister Series of Electrolyte Solutions Alexandre P. dos Santos,† Alexandre Diehl,‡ and Yan Levin*,† †
Instituto de Fı´sica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil, and ‡Departamento de Fı´sica, Instituto de Fı´sica e Matem atica, Universidade Federal de Pelotas, Caixa Postal 354, CEP 96010-900, Pelotas, RS, Brazil Received February 9, 2010. Revised Manuscript Received March 22, 2010
A theory is presented which allows us to accurately calculate the surface tensions and the surface potentials of electrolyte solutions. Both the ionic hydration and the polarizability are taken into account. We find a good correlation between the Jones-Dole viscosity B coefficient and the ionic hydration near the air-water interface. The kosmotropic anions such as fluoride, iodate, sulfate, and carbonate are found to be strongly hydrated and are repelled from the interface. The chaotropic anions such as perchlorate, iodide, chlorate, and bromide are found to be significantly adsorbed to the interface. Chloride and bromate anions become weakly hydrated in the interfacial region. The sequence of surface tensions and surface potentials is found to follow the Hofmeister ordering. The theory quantitatively accounts for the surface tensions of 10 sodium salts for which there is experimental data.
Introduction Electrolyte solutions have been the subject of intense study for over a century. While the bulk properties of electrolytes are now quite well understood, their behavior at interfaces and surfaces still remains a puzzle. Over 100 years ago, Hofmeister observed that presence of electrolyte in water modified significantly the solubility of proteins. While some salts lead to protein precipitation (salting out), other salts stabilize proteins, increasing their solubility (salting in). A few years after this curious observation, Heydweiller1 discovered that salt dissolved in water increased the surface tension of the solution-air interface. While cations had only a small influence on the surface tension, anions affected it quite significantly. Furthermore, the magnitude of the variation of the surface tension followed the same sequence discovered by Hofmeister earlier. It appeared that the two effects were related. Because of its great importance for biology, over the past century there has been a tremendous effort to understand the ionic specificity. Since the air-water interface is relative simple, as compared to proteins, most of the theoretical work has concentrated on it. Langmuir2 was the first to attempt a theoretical explanation of the physical mechanism behind the increase of the surface tension produced by electrolytes. Using the Gibbs adsorption isotherm equation, Langmuir concluded that the phenomenon was a consequence of ion depletion near the air-water interface and suggested that the depleted layer was about 4 A˚ in width. No clear explanation for existence of this ion-free layer was provided by Langmuir, and soon it became clear that in order to obtain a reasonable agreement with experiments, its width had to be a function of ionic concentration.3 A further insight was provided by Wagner,4 based on the Debye-H€uckel (DH) theory of strong electrolytes.5 Wagner argued that ionic depletion was a consequence of the electrostatic repulsion produced by *To whom correspondence should be addressed. E-mail:
[email protected]. (1) Heydweiller, A. Ann. Phys. (Leipzig) 1910, 33, 145. (2) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848. (3) (a) Harkins, W. D.; McLaughlin, H. M. J. Am. Chem. Soc. 1925, 47, 2083. (b) Harkins, W. D.; Gilbert, E. C. J. Am. Chem. Soc. 1926, 48, 604. (4) Wagner, C. Phys. Z. 1924, 25, 474. (5) Debye, P. W.; H€uckel, E. Phys. Z. 1923, 24, 185.
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the interaction of ions with their electrostatic images across the air-water interface. Wagner’s theory was quite complicated, and a simplified version was proposed by Onsager and Samaras (OS), who also derived a limiting law which, they argued, had a universal validity for all electrolytes at sufficiently low concentrations.6 Indeed, careful experimental measurements have confirmed the OS limiting law.7 However at larger concentrations, OS theory was found to strongly underestimate surface tensions. It should be noted, however, that the quantitative measurements of surface tensions are very difficult, and there are some variation in the values reported by different experimental groups.8 Similar to Langmuir, Wagner and OS integrated the Gibbs adsorption isotherm equation to obtain the excess surface tension. Some time ago, Levin and Flores-Mena (LFM) proposed a different approach based on the direct Helmholtz free energy calculation.9 The LFM theory combined Langmuir, Wagner, and OS insights into one theory. They argued that, in addition to the ion-image interaction, ionic hydration leads to a hard-core-like repulsion from the Gibbs dividing surface. Using a 2 A˚ hydrated radius of Naþ and Cl-, they were able to obtain a very good agreement with the experimental measurements of surface tension of NaCl solution, up to 1 M concentration. However, the LFM theory failed to correctly account for the ionic specificity, predicting that the surface tension of NaI solution should be larger than that of NaCl, contrary to experiments. It was clear that the LFM theory was still lacking an important ingredient. An indication of the missing ingredient was already present in the work of Frumkin, 80 years earlier.10 Frumkin measured the electrostatic potential difference across the solution-air interface and found that for all halogen salts, except fluoride, the potential was lower in air than in water. This meant that anions were preferentially solvated near the interface. Bostr€om et al.11 (6) Onsager, L.; Samaras, N. N. T. J. Chem. Phys. 1934, 2, 528. (7) (a) Long, F. A.; Nutting, G. C. J. Am. Chem. Soc. 1942, 64, 2476. (b) Passoth, G. Z. Phys. Chem. (Leipzig) 1959, 211, 129. (8) Henry, C. L.; Dalton, C. N.; Scruton, L.; Craig, V. S. J. J. Phys. Chem. C 2007, 111, 1015. (9) Levin, Y.; Flores-Mena, J. E. Europhys. Lett. 2001, 56, 187. (10) Frumkin, A. Z. Phys. Chem. (Munich) 1924, 109, 34. (11) Bostr€om, M.; Williams, D. R. M.; Ninham, B. W. Langmuir 2001, 17, 4475.
Published on Web 04/02/2010
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suggested that this ionic specificity was a consequence of dispersion forces arising from finite frequency electromagnetic fluctuations.12 Their theory, however, predicted surface potentials of opposite sign to the ones measured by Frumkin, implying that cations were preferentially adsorbed to the interface. This was clearly contradicted by the simulations on small water clusters13 as well as by the subsequent large scale polarizable force fields simulations14 and by the photoelectron emission experiments.15 All these agreed with Frumkin that large anions, and not small cations, are preferentially solvated near the air-water interface. A theoretical explanation for this behavior was advanced by Levin,16 who argued that anionic surface solvation was a consequence of the competition between the cavitational and the electrostatic energies. The cavitational energy arises from the perturbation to the hydrogen bond network produced by ionic solvation. This results in a short-range force which drives ions toward the interface. This is counterbalanced by the electrostatic Born solvation force which arises from the dipolar screening of the ionic self-energy in aqueous environment. For hard (weakly polarizable) ions, the Born energy is much larger than the cavitational energy, favoring the bulk solvation. However for large polarizable ions, the energy balance is shifted. For such ions, ionic charge can easily redistribute itself so that even if a large fraction of ionic volume is exposed to air, the electrostatic energy penalty for this remains small, since most of the ionic charge remains hydrated. This means that through surface solvation large, strongly polarizable ions can have the best of two worlds— gain the cavitational energy at a small price in electrostatic selfenergy. Levin derived the interaction potential quantifying this effect.16 In a follow-up paper, Levin et al.17 used this potential to quantitatively account for the surface tensions of all sodium halide salts. In this paper we will extend the theory of ref 17 to calculate the surface tensions and the surface potentials of others sodium salts and to derive the Hofmeister series.
Model and Theory Consider an electrolyte solution confined to a mesoscopic drop of water of radius R, which corresponds to the position of the Gibbs dividing surface (GDS).17,18 We define the adsorption (ion excess per unit area) as Γ(
1 ¼ 4πR2
"Z 0
¥
4πR3 F ( ðrÞ4πr dr cb 3 2
# ð1Þ
where F((r) are the ionic density profiles and cb = Fþ(0) = F-(0) is the bulk concentration of electrolyte. If N ion pairs are inside the drop, eq 1 simplifies to Γ( = N/4πR2 - cbR/3. The water and air will be treated as uniform dielectrics of permittivities εw = 80 and εa = 1, respectively, with a discontinuity across the GDS. Clearly this is an approximation, the (12) Manciu, M.; Ruckenstein, E. J. Phys. Chem. B 2004, 108, 20479. (13) (a) Perera, L.; Berkowitz, M. L. J. Chem. Phys. 1991, 95, 1954. (b) Dang, L. X.; Smith, D. E. J. Chem. Phys. 1993, 99, 6950. (c) Stuart, S. J.; Berne, B. J. J. Phys. Chem. A 1999, 103, 10300. (14) (a) Jungwirth, P.; Tobias, D. J. J. Phys. Chem. B 2002, 106, 6361. (b) Jungwirth, P.; Tobias, D. J. Chem. Rev. 2006, 106, 1259. (c) Horinek, D.; Herz, A.; Vrbka, L.; Sedlmeier, F.; Mamatkulov, S. I.; Netz, R. R. Chem. Phys. Lett. 2009, 479, 173. (d) Brown, M. A.; D'Auria, R.; Kuo, I. F. W.; Krisch, M. J.; Starr, D. E.; Bluhm, H.; Tobias, D. J.; Hemminger, J. C. Phys. Chem. Chem. Phys. 2008, 10, 4778. (15) (a) Markovich, G.; Pollack, S.; Giniger, R.; Cheshnovski, O. J. Chem. Phys. 1991, 95, 9416. (b) Ghosal, S.; Hemminger, J.; Bluhm, H.; Mun, B.; Hebenstreit, E. L. D.; Ketteler, G.; Ogletree, D. F.; Requejo, F. G.; Salmeron, M. Science 2005, 307, 563. (c) Garrett, B. Science 2004, 303, 1146. (16) Levin, Y. Phys. Rev. Lett. 2009, 102, 147803. (17) Levin, Y.; dos Santos, A. P.; Diehl, A. Phys. Rev. Lett. 2009, 103, 257802. (18) Ho, C. H.; Tsao, H. K.; Sheng, Y. J. J. Chem. Phys. 2003, 119, 2369.
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validity of which can only be tested a posteriori. We note, however, that at any given instant an ion adsorbed to the interface will see it as almost flat—i.e. interface fluctuates, but the ion moves along with it. The dielectric environment seen by the ion at any instant should, therefore, be well approximated by a jump in the dielectric constant. The surface tension can be obtained by integrating the Gibbs adsorption isotherm equation, dγ = -Γþ dμþ - Γ- dμ-, where μ( are the chemical potentials of cations and anions, respectively. Let us first consider alkali-metal cations, such as lithium, sodium, or potassium. Because these cations are small, they have large surface charge density, which leads to strong interaction with surrounding water molecules, resulting in an effective hydrated radius ah. We can, therefore, model these ions as hard spheres of radius ah with a point charge q located at the origin. Because of their strong hydration, these cations cannot move across the GDS since this would require them to shed their solvation sheath. For mesoscopic drops we can neglect the curvature of the GDS. To bring a cation from bulk electrolyte to some distance z > ah from the GDS requires9,17 Wðz; ah Þ ¼
q2 2εw
Z
¥
dk e - 2sðz - ah Þ
0
k½s coshðkah Þ - k sinhðkah Þ s½s coshðkah Þ þ k sinhðkah Þ ð2Þ
of work. The GDS is located at z = 0, and the axis is oriented into the drop. We have defined s = (κ2 þ k2)1/2, where κ = (8πq2cb/ εwkBT)1/2 is the inverse Debye length. Equation 2 is well approximated by Wap ðz; ah Þ ¼
Wðah ; ah Þah - 2Kðz - ah Þ e z
ð3Þ
(Supporting Information, Figure S1). This form will be used later to speed up the numerical calculations. Unlike small cations, large halogen anions of bare radius a0 have low electronic charge density and are weakly hydrated. The polarizability of an anion is γa, and we define its relative polarizability as R = γa/a03. We can model these ions as imperfect spherical conductors. When an anion is far from the interface, its charge -q is uniformly distributed over its surface. However, when an anion begins to cross the GDS, its charge starts to redistribute itself on the surface so as to leave most of it in the high dielectric environment.16 The fraction of charge x which remains hydrated when the ionic center is at distance z from the GDS is determined by the minimization of the polarization energy16 " # q2 πx2 π½1 - x2 εw g 1 - cos½θðzÞ 2 Up ðz, xÞ ¼ þ xþ β 2 2a0 εw θðzÞ ½π - θðzÞε0 ð4Þ where θ(z) = arccos[-z/a0] and g = (1 - R)/R. We find xðzÞ ¼
λB πεw λB π λB πεw þ g½1 - cos½θðzÞ = þ þ 2g a0 ε0 ½π - θðzÞ a0 θðzÞ a0 ε0 ½π - θðzÞ
ð5Þ Substituting this back into eq 4, we obtain the polarization potential. This potential is repulsive, favoring ions to move toward the bulk. Nevertheless, the repulsion is quite soft compared to a hard-core-like repulsion of strongly hydrated cations. The force that drives anions toward the interface arises from cavitation. When ion is dissolved in water, it creates a cavity from DOI: 10.1021/la100604k
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which water molecules are expelled. This leads to a perturbation to the hydrogen bond network and an energetic cost. For small voids, the cavitational energy scales with their volume.19 As ion moves across the interface, the cavity that it creates in water diminishes proportionally to the fraction of the volume exposed to air,16,17 producing a short-range interaction potential that forces it to move across the GDS 8 < νa0 3 for zga0 2 Ucav ðzÞ ¼ 1 3 z z : νa0 for - a0 < z < a0 þ1 24 a0 a0 ð6Þ where ν ≈ 0.3kBT/A˚3 is obtained from bulk simulations.20 For small strongly hydrated cations, this energy gain does not compensate for the electrostatic energy penalty of exposing the ionic charge to the low dielectric environment. For large polarizable ions, on the other hand, electrostatic energy penalty is small, and the cavitational energy is sufficient to favor the surface solvation. The total potential felt by an unhydrated anion is then17 8 q2 > > 3 > for zga0 < Wðz; a0 Þ þ νa0 þ 2ε a w 0 Utot ðzÞ ¼ > Wða0 ; a0 Þz=a0 þ Up ðzÞ þ Ucav ðzÞ for 0 < z < a0 > > : Up ðzÞ þ Ucav ðzÞ for - a0 < ze0 ð7Þ The density profiles can now be calculated by integrating the nonlinear modified Poisson-Boltzmann equation (mPB): 4πq ½F ðrÞ - F - ðrÞ εw þ NΘðR - ah - rÞe - βqφðrÞ - βWðz;ah Þ F þ ðrÞ ¼ R R - ah 4πr2 dr e - βqφðrÞ - βWðz;ah Þ 0 r2 φðrÞ ¼ -
F - ðrÞ ¼ R R þ a0 0
ð8Þ
NeβqφðrÞ - βUtot ðrÞ 4πr2 dr eβqφðrÞ - βUtot ðrÞ
where β = 1/kBT and Θ is the Heaviside step function. The boundary condition for this mPB equation is vanishing of the electrostatic potential and of electric field at R þ a0. To speed up the numerical calculations, we can replace W f Wap.
Figure 1. Excess surface tensions of NaF, NaCl, NaBr, and NaI solutions. The symbols are the experimental data,23-25 and the lines are the results of the present theory. The inset shows the surface potential difference as a function of molar concentration.
this work are nonperturbative, there is no need to artificially lower the polarizabilities. The excess surface tension can be obtained by integrating the Gibbs adsorption isotherm equation (eq 1). At the level of PB approximation the chemical potential that enters into equation (eq 1) is simply that of an ideal gas, βμ( = ln(cbΛ(3), where Λ( are the de Broglie thermal wavelengths. Comparing to more sophisticated theories and simulations, we find that this approximation works quite well up to fairly large concentrations ≈1 M.9,18 We start with NaI. Since I- is large and soft, it should be unhydrated in the interfacial region. Adjusting the hydrated radius of Naþ to best fit the experimental data23 for NaI, we obtain17 ah,Na = 2.5 A˚. We will use this partially hydrated radius of Naþ in the rest of the paper. Considering that Br- is also large and soft, we expect that it will also remain unhydrated in the interfacial region. This expectation is well justified, and we obtain a very good agreement with the experimental data25 (see Figure 1). For F- the situation should be very different. This ion is small, hard, and strongly hydrated. This means that, just like for a cation, a hard-core repulsion from the GDS must be explicitly included in the mPB equation. For hydrated (or partially hydrated) anions the density is then
Sodium-Halogen Salts
NΘðR - ah - rÞeβqφðrÞ - βWðz;ah Þ F - ðrÞ ¼ R R - ah 4πr2 dr eβqφðrÞ - βWðz;ah Þ 0
First, we study the sodium-halogen salts.17 The anion radii were obtained by Latimer, Pitzer, and Slansky21 by fitting the experimentally measured free energies of hydration to the Born model. Since our theory in the bulk also reduces to the Born model, these radii are particularly appropriate: aI = 2.26 A˚, aBr = 2.05 A˚, aCl = 1.91 A˚, and aF = 1.46 A˚. For ionic polarizabilities we use the values from ref 22: γI = 7.4 A˚3, γBr = 5.07 A˚3, γCl = 3.77 A˚3, and γF = 1.31 A˚3. We should note that in simulations the values of ionic polarizability are sometimes artificially lowered to avoid the polarization catastrophe. This catastrophe is a consequence of the truncation of the multipole expansion at the dipolar level. Since the calculations presented in
In this case, the mPB boundary condition becomes vanishing of the potential and of electric field at r = R - ah,
